Coresets for Robust Clustering via Black-box Reductions to Vanilla Case

TL;DR

This paper introduces a method to create small, robust coresets for clustering with outliers by reducing the problem to the vanilla case, enabling efficient algorithms in various metric spaces and streaming settings.

Contribution

It presents a black-box reduction technique to derive robust coresets from vanilla coresets, improving size bounds and extending applicability to streaming algorithms.

Findings

Coresets of near-linear size for robust clustering with outliers.

First streaming algorithms for $k$-Median and $k$-Means with outliers.

New theoretical conditions under which vanilla coresets are also robust coresets.

Abstract

We devise $\epsilon$-coresets for robust $(k,z)$-Clustering with $m$ outliers through black-box reductions to vanilla case. Given an $\epsilon$-coreset construction for vanilla clustering with size $N$, we construct coresets of size $N\cdot \mathrm{poly}\log(km\epsilon^{-1}) + O_z\left(\min\{km\epsilon^{-1}, m\epsilon^{-2z}\log^z(km\epsilon^{-1}) \}\right)$ for various metric spaces, where $O_z$ hides $2^{O(z\log z)}$ factors. This increases the size of the vanilla coreset by a small multiplicative factor of $\mathrm{poly}\log(km\epsilon^{-1})$, and the additive term is up to a $(\epsilon^{-1}\log (km))^{O(z)}$ factor to the size of the optimal robust coreset. Plugging in vanilla coreset results of [Cohen-Addad et al., STOC'21], we obtain the first coresets for $(k,z)$-Clustering with $m$ outliers with size near-linear in $k$ while previous results have size at least $\Omega(k^2)$ [Huang et al., ICLR'23; Huang et al., SODA'25]. Technically, we establish two conditions under which a vanilla coreset is as well a robust coreset. The first condition requires the dataset to satisfy special structures - it can be broken into "dense" parts with bounded diameter. We combine this with a new bounded-diameter decomposition that has only $O_z(km \epsilon^{-1})$ non-dense points to obtain the $O_z(km \epsilon^{-1})$ additive bound. Another condition requires the vanilla coreset to possess an extra size-preserving property. We further give a black-box reduction that turns a vanilla coreset to the one satisfying the said size-preserving property, leading to the alternative $O_z(m\epsilon^{-2z}\log^{z}(km\epsilon^{-1}))$ additive bound. We also implement our reductions in the dynamic streaming setting and obtain the first streaming algorithms for $k$-Median and $k$-Means with $m$ outliers, using space $\tilde{O}(k+m)\cdot\mathrm{poly}(d\epsilon^{-1}\log\Delta)$ for inputs on the grid $[\Delta]^d$.